Great circle computer



Feb. 12, 1957 I. A. GREENWOOD, JR 2,781,168

GREAT CIRCLE COMPUTER 2 Sheets-Sheet 1 Filed Sept. 15, 1955 'INVENTOR.m9 4. aeawnvoo, .12.

1957 l. A. GREENWOOD, JR 2,781,168

GREAT CIRCLE COMPUTER 2 Sheets-Sheet 2 Filed Sept. 15. 1955 '1 ms 1 so:

IN V EN TOR. H4411 ,4. G'Eff/VWOOQ/ 7/ 77/4 United States Patent GREATCIRCLE COMPUTER Ivan A. Greenwood, J12, Stamford, Conn., assignor toGeneral Precision Laboratory Incorporated, a corporation of New YorkApplication September 15, 1955, Serial No. 534,578

9 Claims. (Cl. 235-61) This invention pertains to a computer for solvingspher ical triangles and is particularly useful in navigating anyvehicle relative to the earths surface. In such instances the computeris utilized to determine the course angle for following a great circleroute and the distance to destination.

A basic problem in navigation involves the computation of the greatcircle course between two points on the earths surface. Such a coursefollows the shortest path between the two points but in following it thecourse angle continuously changes. in the case of ocean vesselnavigation the course angle is frequently recomputed and reset; thecourse is then a series of chords joining the various points on theapproximate path of the great circle route. In the case of airnavigation recomputation and resetting of the course must be morefrequent because of the greater speed, and it is highly preferable tomake automatic and continuous recomputations.

The computer of the present invention solves automatically for bothcourse angle and distance to destination so that if continuouslycorrected data are presented to the computer the output of course anddistance to destination will be continuously indicated. The initialinput data for this computer consists of the latitudes and longitudes ofthe destination and of the starting point. When the position of thedestination is fixed the only data to be furnished to the iustrmnentduring the journey are the.

continuous changes in the instant latitude and longitude of the vehicle.However, should the destination be altered after the vehicle has leftthe original fixed point of departure the present invention is capableof responding to new input data consisting of the latitude and longitudeof the new destination, therefrom computing a new great circle routefrom the instant position of the vehicle to the new destination. inother words, when data are set into the device corresponding to twopoints on the earths surface, one of which is the present position ofthe vehicle, the device will instantly compute the course angle for thegreat circle route to the destination and the exact distance to betravelled.

The device will continue to recompute these quantities so that the dialsshowing them are kept accurate and up to date from instant to instant.

A difiiculty which has been encountered in devices of this nature andwhich has not been overcome so far as known, is that these devicesrequire a multiplicity of components each of which has instrument errorso that in the aggregate a considerable error may accrue in the courseangle and in the computed distance to destination.

eifacing with time so that as the destination is neared the instrumenterrors tend to vanish and at the destination resulting in thetheoretically shortest route.

2,781,168 Patented Feb. 12, 1957 "ice do vanish completely. Thus avehicle equipped in accordance with this invention homes on itsdestination with complete elimination of all instrument error.

The manner of securing the input data is outside of the scope of thisinvention. They may be secured by frequent or continuous celestialobservation, by manual or automatic dead reckoning means based oncompass and ground speed data, by radar or other electronic navigationaids or in any other way.

In the instant invention, representations of latitude and longitudeconstituting the current input data are continuously presented to thecomputer in the form of mechanical shaft displacements. By means ofcomponents which resolve, multiply, add, subtract and solve triangles,two output shafts are continuously repositioned, one representing thecourse to be steered in terms of azimuth angle and the otherrepresenting the distance remaining to destination by great circleroute.

The general purpose of this invention then is to provide a great circlenavigation computer for continuously indicating the course and greatcircle distance to destination.

The particular purpose of this invention is to provide such a computerin which the instrument error grows less as the objective is approachedand completely vanishes as the objective is reached.

A further understanding of this invention may be secured from thedetailed description and accompanying drawings, in which:

Figure 1 represents a spherical triangle drawn on the surface of theearth, for use in explaining the navigational equations underlying thetheory of this invention.

Figure 2 is a schematic drawing of the circuit of an instrumentembodying this invention.

It has been discovered that when the following two equations are used asbasis for construction of a great circle computer, the instrument errorcontinuously grows less and becomes zero at destination.

sin D sin CA=COS LB sin L0 These two trigonometric identities describeproperties of spherical triangles and are derived from the laws of sinesand cosines by trigonometric manipulation. They are applied to aspherical triangle for use in navigation as shown in Fig. l, in whichthe circle 11 represents the earth, with the Equator at 12 and the NorthPole at 13. A and B represent any two points on the earths surface, Abeing the start of a navigational path or the present position of anavigated vehicle, and B the termination of the path.

Navigation is to be executed by the great circle method, This route isindicated at 14 and has a length in units of are termed D. The meridiansthrough A and B are designated 16 and 17 and these meridians togetherwith the great circle path 14 form a spherical triangle drawn on theearths surface having as apices the North Pole and points A and B. Thelongitude of A and B are Lon and Lon, respectively, and the longitudedifference is L0, or

The latitudes of A and B are LA and LB, respectively, and their latitudedifference is L, or

The angle between the sides 14 and 16 of the spherical triangle isdesignated CA and is the course angle at A in terms of azimuth degrees.

Referring now to Fig. 2, the latitude LB and longitude Lon of thedestination are set into the instrument by shafts 21 and 22respectively. Starting position or present position data in terms oflatitude LA and longitude LoA nventors are supplied to the instrumentthrough shafts-23 anrl-24 respectively, the angular displacements ofthese shafts representing the magnitudes of these quantities in angular'terms. The longitudes Lon and Lon are subtracted 'in device- 26,whi'chmay for example be a spur "gear dif ferential, toform thelongitude difference angle L at shaft '27. Similarly, the latitudes LAand LB are subtracted in device '28 to form the latitude differenceangle L at shaft 29. 'This permits the differences to be formed to highdegrees of accuracy, even though the input data contain systematicerrors.

The four quantities L, La, LA and LB are how separately andtrigonometrically resolved into their sine or cosine functions bysynchro resolvers of conventional form. For example, the shaft 29,representing by' its angulardisplacement the latitude difference angleL, is connected to therefor winding -31 of a synchro resolver 32.

A stator winding 33 is excited by alternating current applied atterminal 34 so that the output potential at con: ductor 36 is related tothe stator potential in terms of the sine or cosine, in this case thesine, of the rotor angle.

' The potential of conductor 36 therefore is representative of sin L. Ip g 'In similar manner the synchro resolvers 37, 38 and 39 have outputsat conductors 41, 42 and 43 representing by 'their potential magnitudessin Lo, sin LA, and cos LB,

respectively. 7

The output conductor 43 hearing a potential representing cos Ln isconnected to a position servomechanism 44 of conventional formcomprising subtracting resistors 46 and 47, amplifier 48, motor 49, andlinear voltage divider 51. Bymeans of this servomechanism the potentialment of an output shaft 52. This shaft displacement thus is proportionalto and represents cos LB.

This shaft displacement is applied to displace the slider -53 of alinear voltage divider 54, the divider being con- -nected to conductor41. The electrical output at slider 53 is'therefore' the product of thevoltage divider'excitation, sin Lo, and its-slider position, cos Ln,andthus represents the terrn cos Ln sin L0. This term is identical withthe right side of Equation 1.

In similar manner a linear voltage divider-'56 is connected to conductor42 for excitation by a potential equal to sin Ligand the slider 57 isconnected for positioning lay-shaft 52 in accordance with the functioncos LB. The slider potential then represents the product or cos LB vAnon-linear voltage divider 58 is designed to produce an output voltage yhaving the form 7 when electrically excited by a voltageE andmechanically moved in accordance with the quantity Lo.- Since it isactually' electrically excited by a quantity representing cos Ls sin LA,Equation 5 becomes:

y =cos VLB sin LA(1COS L0) This'quantity is represented by the outputpotential at conductor 61 connected to the voltage divider slider 59.

This quantity is identical with the last term of Equation 2. It isessential that this function can actually go to zero as instrumented,therefore the divider 58 should be adjusted to" have zero output whenthe L0 input is zero.

The terms sin L represented 'by the potential of conductor 36, and cosLB sin LA, (I -cos Lo) represented by 4 i --M- sin N=e1 (7) M c'os N=e2when (21 and 22 are applied as input data. its operation is wellunderstood in the art. It contains a double resolver having inputst-atorcoils es and s9 and output rotor coils 71 and 72. Coil 7Z'is connectedthrough servo amplifier 73 and motor 74 to position the rotor shaft into the null position of winding '72.

of conductor 77 will represent M. The'left sides of Equations 7 and 8are in the same form as those of Equations 1 and 2, so that if M'=sin D(9) and N =CA (10) then e1=c0s LB sin L0 (11) and 7 e2=sin L-l-cos LBsin LA(1COS L0) (12) Since these quantities equal to m and 22 areapplied through conductors 66 and 78 to input stator windings 68 and 69,it follows that the output shaft '76 is positioned in accordance withthe angle CA and the output conductor 77 bears a potential whose valueis representative of sin D.

The potential representing sin D'is converted into a shaft displacementrepresenting D by means of a conventional position servomechanism 79comprising an addingservo amplifier 81, motor 82, and syuchro resolverof' conductor 43- is'represented by the angular displace- 83. The shaft84 of the synchro resolver 33 is positioned in accordance with theangular measure of D. V V

, The shafts 75 and 34, representing by their angular deflections thequantities CA and D, are connected to dials 86 and 87, from which thesequantities may be read at any time and the vehicle navigated inaccordance with them.

' It is obvious that for the purpose of steering the vehicle the angleCA only is necessary and the quantity D is not needed. For this purpose,therefore, the dial 87 V and position servomechanism 79 can be omitted.However, the quantity D is generally necessary because it indicatesarrival at the destination, so that, for a self-contained navigationsystem not relying on visual or other aids for this information, D is asnecessary as the quantity CA in navigation to and not beyond an exactdestination point B. In addition, D is useful during the journey inpredicting time of arrival at destination.

when L0 is zero.

The resistors 88 and 89 are used to match the resistance of voltagedivider 58. Since the latter is bridged to the slider 57 of voltagedivider 56, it destroys the linearity of divider 56 and results in acurved characteristicat "slider 57. When matching curved characteristicsare given to the dividers 51 and 54 by means of the resistors 88 and 89,inaccuracy due to this cause is eliminated.

The instrument of this invention has complete absence of instrumenterror at the destination because each instrumented term reduces to zeroat the destination and because all instrument errors are adjusted toneutralization at the destination. That is, reliance is not placed onsubtraction of two large terms which at destination are equalexcept fortheir contained instrument errors, which in general will, of course, notbe equal and will not cancel" each other. 7 Referring to theinstrumented right-hand sides of Equations 1 and 2, cos Ln sin Lobecomes zeroat destination-because L0 is zero; sin L becomes zerobecause L is zero; and cos LB sin LA (1-cos L0) becomes zero because theparenthetical term is zero Instrument errors are adjusted toneutralization in the following manner. hat theerror of the output ofservo "44'be +k, so that for a selected-value of LB the output. 1 is cosLn -i-k. 1 Similarly, let the error of the output of synchro 37 be m sothat at a selected value of L0 the' The'angle of shaft 76 willthen'represen'tthe angle N and the-potential output output is sin Lo+m.Howevenm is reduced to zero at destination by adjustment of 37 with aninput Lo=0. Let the error added by voltage divider 54 at a selectedinput be n. Its output will then be (cos LB-j-k) sin Lo-l-n (11) andwhen L is zero the first term vanishes. The voltage divider 54 is nowadjusted so that when L0 is zero, the divider output is zero, thusmaking n equal to zero at this specific point. As the result, thefunction cos LB sin L0 fed through conductor 78 from slider 53 towinding 69 of arc tangent solver 67 may include instrument errors whileapproaching the destination, but when near to the destination theseerrors will be small and at the destination there will be no error andthe term cos LB sin Lo will be precisely zero.

A similar procedure is followed in the elimination of error ininstrumentation of the right side of Equation 2.

Let the error in resolver 32 be termed p, so that its output in generalis not sin L, but sin L+p. This resolver is now adjusted so that, whenL=0, p=0. This eliminates all error in the output of resolver 32 atdestination.

As respects the second term of Equation 2, cos LB sin LA (1cos L0), iffortuitous instrument errors added at a selected value to the outputs ofservo mechanism 44, resolver 33, and dividers 56 and 58 are k, q, r ands, respectively, the voltage output in conductor 61 will be cos LB sinLA (l-cos L0)+(1cos L0) [1 (cos LB, sin LA, kq, r)]-j-s [f (cos LB, sinLA, kq, r)] (12) However, the versed sine (l-cos L0) approaches zeromuch more rapidly than L0 or L, as the destination is neared, so thatthe first two terms of this expression are small and the second or errorterm can be neglected as undetectible in practice. The third term ismade to vanish by adjusting inputs to make Lo zero, then adjusting thevoltage divider 58 to make its output zero under this condition. Thishas the efiect of giving 5 such value as to neutralize kg and r. Theterm cos LB sin LA (1-cos L0) is thus made to have little or no erroradded to it at destination, and since its magnitude near destination isonly a very small fraction of the magnitude of the other term ofEquation 2, i. e., sin L, the error is completely negligible.

The instrumentation depicted in Fig. 2 indicates connections at severalpoints to a source of electrical power. Fluctuations in line voltage ofthis source are, however, so balanced as to have no effect upon theoutput data accuracy because in efiect line voltage variations appear onboth sides of an equation, or in both numerator and denominator of afraction.

'i his instrument thus provides indications of its two output dials ofthe quantities D and LA which do not cont :1 input data errors becausethese errors cancel in the difierence terms LOA-LOB and LA-LB.Additionally instrument errors vanish at destination and areincreasingly minimized during the journey.

What is claimed is:

l. A course angle computer for the great circle navigation of a vehiclefrom any present position to a predetermined destination comprising,means for introducing latitudes and longitudes of present position anddestination, first means adapted for reception of present latitude anddestination latitude data having an output representative of the sine ofthe latitude difference, second means adapted for reception of presentposition and destination latitude and longitude data to produce anoutput representative of the cosine of destination latitude multipliedby sine of present latitude multiplied by the versed sine of longitudediiference, third means adapted for reception of destination latitudeand longitude and present position longitude to produce an outputrepresentative of the product of the cosine of destination latitude andthe sine of the longitude ditference, summation means for '8 adding theoutputs of said first and second means, and are tangent means actuatedby said summation means and said third means to produce a shaft angleequal to said course angle.

2. A course angle computer in accordance with claim 1 including an arctangent means electrical output representing the sine of the presentdistance between present position and destination.

3. A course angle and distance to destination computer in accordancewith claim 2 including arc sine resolver means connected for actuationby said arc tangent means electrical output to form an outputrepresenting the distance between present position and destination.

4. A course angle computer for the great circle navigation of a vehiclefrom any present position to a predetermined destination comprising,input means for introducing the latitudes and longitudes of'presentposition and destination, first subtracting means for subtracting saidlatitudes to produce a latitude difierence indication, secondsubtracting means for subtracting said longitudes to produce a longitudedifierence indication, servo means actuated by said destination latitudeinput for deflecting a shaft in accordance with the cosine ofdestination latitude, first resolving means actuated by said latitudesubtracting means for producing a signal representative of the sine ofthe latitude difference, second resolving means actuated by said presentlatitude input for producing a signal representative of the sine ofpresent latitude, third resolving means actuated by said longitudesubtracting means for producing a signal representative of the sine ofthe longitude difierence, first multiplying means actuated by said servomeans and said second resolving means to produce a signal representativeof the product of the cosine of destination latitude and the sine ofpresent latitude, second non-linear multiplying means actuated by saidsecond subtracting means and said first multiplying means to produce asignal representative of the product of the cosine of destinationlatitude, sine of the present latitude and the versed sine of longitudedifference, third multiplying means actuated by said third resolvingmeans and said servo means to produce a signal representative of theproduct of the cosine of destination latitude and the sine of thelongitude difference as a first component voltage, adding meansconnected to said first resolving means and second non-linearmultiplying means to form a second component voltage, and vector addingmeans connected to said third multiplying means and said adding means toform the vector sum angle of said first and second component voltages,said vector sum angle therefore being said course angle.

5. A course angle computer in accordance with claim 4 in which saidvector adding means includes a voltage out put representative of themagnitude of said vector sum and therefore representative of the sine ofthe distance between present position and destination.

6. A course angle computer in accordance with claim 5 includingservomechanism arc sine means connected to said vector adding meanshaving a mechanical shaft deflection output representative of themagnitude of the arc sine of said vector sum and thereforerepresentative of the distance between present position and destination.

7. A computer for electrically computing the great circle course anglefor navigating a vehicle from any present position to a predetermineddestination comprising, a latitude subtracting differential having twoinput shafts angularly adjusted to present latitude and destinationlatitude and having an output shaft, a longitude subtractingdifferential having two input shafts angularly adjusted to presentlongitude and destination longitude and having an output shaft, a firstsynchro resolver having its shaft angularly adjusted to destinationlatitude generating an electrical output quantity representative of thecosine of destination latitude, a sei'vomechanism connected to saidfirst synchro resolver having a shaft output representative of thecosine of destination latitude,

' energization 'by the output of said first a second -synchroresolverhaving its shaft connected tor actuationbydheoutput shaft of thelatitude subtracting difirential generating-an electrical outputquantity represynchro resolver having its shaft connected for actuationby the output shaft of the longitude subtracting differential togenerate an electricaloutput quantity representative of the sine of thelongitude difierence; a first linear voltage divider electricallyenergized by the output of said third synchro resolver and mechanicallyconnected for slidenactuation by theshait output of said servomechanismandhaving an electrical slider output, a second non-linear voltagedivider connected for electrical elt' e divider and'mechanicallyconnected for slider actuation by the output shaft of said? longitudesubtracting differential and having .anelectrical slider output, a thirdlinear voltageidivider electrically connected for'energization by theoutput of said fourth synchro resolver and mechanically connected forslider actuation by the shaft output of said servomechanism and havingan electrical-slider output, an addingudevice connected for energizationby the outputs of said second "syuchroTesol-veuandof said secondaion--linear voltage di ider-totem an'el'ectric'al 1 output' repre senting--the "sum thereof, and an arc "tangent "solver "con-: nected foractuation bytlre =output voltages of said adding device and of saidthirdlinearcvoltage divider and hav ing a shaft output angle whosetangent is the ratio of said said are tangenfsolverhas an electricaloutput representing the vector sum of said two actuating voltages and tnumerically equal to the sine of the distance between said presentposition and said predetermined destination.

9. A computer in accordance with claim 8 including a secondservomechanism connected for actuation by the electrical output of saidare tangent solver and having a mechanical shaft output deflectionnumerically representing the distance between said present position andsaid predetermined destination.

- References Citedin the file of this patent UNITED STATES PATENTS2,688,440 Gray et a1. Sept. 7; 71954

